Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Entanglement in 2 p atom pairs in H 2 photodissociation as studied by measuring the angular correlation function of Lyman- α photon pairs on a whole sphere

Yutaro Torizuka, Kouichi Hosaka, Philipp Schmidt, Takeshi Odagiri, Arno Ehresmann, M. Kitajima, Noriyuki Kouchi

Published Feb 21, 2025• 1 citation
Publisher

Abstract

An angular correlation function (ACF) of a pair of Lyman-$\ensuremath{\alpha}$ photons produced in photodissociation of ${\mathrm{H}}_{2}$ with linearly polarized incident light is measured on a whole sphere at a 33.66-eV photon energy so that we may distinctly substantiate that the $2p$ atom pairs are in the state previously identified by Torizuka et al. as the $2p$ atom-pair state [Phys. Rev. A 99, 063426 (2019)], who measured the ACF only on a circle in the plane perpendicular to the incident light beam. The $2p$ atom-pair state determined with much less uncertainty than before is investigated in terms of whether or not the state is entangled. We discover that the pair of fragment $\mathrm{H}(2p)$ atoms is certainly entangled in terms of the electron-electron entanglement, and obtain the entanglement entropy of the $2p$ atom-pair state. The similar discussion is made in terms of the electronic spatial-spin entanglement too, where the spatial motion and spin motion of electrons are considered nonidentical partial systems at variance with the former electron-electron entanglement. It turns out that the pair of fragment H($2p$) atoms is entangled in terms of the electronic spatial-spin entanglement as well as the electron-electron entanglement.

DOI
10.1103/physreva.111.023116

Related papers

Entangled pairs of 2p atoms produced in photodissociation of H2 and D2

Yutaro Torizuka, Kouichi Hosaka, Philipp Schmidt +6 more

The angular correlation functions (ACFs) of a pair of Lyman-$\ensuremath{\alpha}$ photons in photodissociation of ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ are measured with linearly polarized incident light at a 33.66-eV incident photon energy in a wider angular range at a narrower angular step with smaller distortion than before [Y. Nakanishi et al., Phys. Rev. A 90, 043405 (2014)] so that we identify the atom-pair state emitting the pair of Lyman-$\ensuremath{\alpha}$ photons and find out whether the atom pair is entangled or not. Searching for reasonable $2p$ atom-pair states that reproduce the experimental ACFs to solve the issue, we show that hydrogen molecules are photoexcited to the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state in the Franck-Condon region and then the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state comes to superpose with the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}(2)$ state as the internuclear distance increases to infinity. The superposition is brought about by the spin-orbit coupling, which is effective around infinite internuclear distance because the potential-energy curves of those states are close to each other, but is negligibly small around the Franck-Condon region because they are apart from each other. The $2p$ atom pairs turn out to be in the ${1}_{u}$ superposition state, which is entangled. We therefore conclude that an entangled pair of hydrogen atoms is spontaneously produced through the photodissociation of a hydrogen molecule, and the entanglement originates from the ${1}_{u}$ symmetry properties, which are invariant during the dissociation from the Franck-Condon region towards infinite internuclear distance.

Physical Review A•Jun 25, 2019•7 citations

Angular correlation of a pair of Lyman-α photons produced in the photodissociation of H2

Yuko Nakanishi, Kouichi Hosaka, 陵子 向後 +6 more

The angular correlation functions (ACFs) of a pair of Lyman-\ensuremath{\alpha} photons emitted from H fragments in the photodissociation of a hydrogen molecule are measured at a 33.66-eV incident photon energy and at hydrogen gas pressures of approximately 0.1 and 1 Pa. The ACFs are measured for both opposite and nonopposite arrangements of the two photon detectors. It turns out that the experimental ACFs involve neither the contribution of the reactions $\mathrm{H}(n\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}2)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{2}$ nor the contribution of the cascade from $\mathrm{H}(n$ \ensuremath{\geqslant} 3) to H($2p)$ fragments. Thus the experimental ACFs are those for primary H($2p)$ pairs following the photodissociation of ${\mathrm{H}}_{2}$. The experimental ACFs are compared with (i) the theoretical ACF for entangled pairs of H($2p)$ atoms, where the magnetic quantum number of each hydrogen atom is indefinite, and (ii) the theoretical ACF for H($2p)$ pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis [the former entangled state of H($2p)$ pairs is a sum of the latter pair states with definite magnetic quantum number]. In the theoretical ACF in (ii), the disentanglement in H($2p)$ pairs during the dissociation is considered. The experimental ACFs show a similar tendency in angular dependence to the theoretical ACF for entangled H($2p)$ pairs. However, there still remains a considerable difference in the variation magnitude between those experimental and theoretical ACFs. The experimental ACFs show the reverse tendency in angular dependence to the theoretical ACF for H($2p)$ pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis. We thus conclude that the pair of H($2p)$ atoms in the photodissociation of ${\mathrm{H}}_{2}$ is unlikely to be in the definite states of magnetic quantum number of each hydrogen atom relative to the internuclear axis, i.e., unlikely to be in the components of the entangled state of H($2p)$ pairs.

Physical Review A•Oct 8, 2014•12 citations

Dynamics of entangled H(2p) pair generated in the photodissociation of H2

Takeshi Odagiri, Takehiko Tanabe, Noriyuki Kouchi

We have measured the coincidence time spectra of two Lyman-α photons emitted by a pair of H(2p) atoms in the photodissociation of H 2 at the incident photon energy of 33.66 eV and at the hydrogen gas pressures of 0.40 Pa and 0.02 Pa, from which the angular distributions of two Lyman-α photons have been obtained. The experimental angular distributions indicate the generation of the entangled pair of H(2p) atoms as predicted by the theory of our group (Miyagi et al 2007 J. Phys. B 40 617) and the role of a new kind of reaction, i.e. the reaction of the entangled pair of H(2p) atoms with an H 2 molecule that efficiently changes the entanglement. It has turned out that more entangled pairs of H(2p) atoms survive at 0.02 Pa than at 0.40 Pa. The decay time constant in the coincidence time spectrum at 0.02 Pa is approximately half the lifetime of a single H(2p) atom, 1.60 ns, while the decay time constant at 0.40 Pa is in agreement with the lifetime of a single H(2p) atom. It follows that the decay faster than the lifetime of a single H(2p) atom originates from the entanglement in the pair of H(2p) atoms.

Journal of Physics Conference Series•Nov 5, 2012•5 citations

Large Pressure Effect on the Angular Distribution of Two Lyman- α Photons Emitted by an Entangled Pair of H(2p) Atoms in the Photodissociation of H2

Takehiko Tanabe, Takeshi Odagiri, M. Nakano +2 more

The angular distribution of two Lyman-alpha photons, i.e., the probability density that two Lyman-alpha photons are emitted in given directions, in the photodissociation of a hydrogen molecule have been measured at the hydrogen gas pressures of 0.40 and 0.13 Pa. We have found that the experimental angular distributions seem to approach the theoretical one by our group [J. Phys. B 40, 617 (2007)] with decreasing pressure, which indicates the generation of the entangled pair of H(2p) atoms shown in the theory and the role of the reaction of the entangled pair of H(2p) atoms with an H2 molecule that efficiently changes the entanglement.

Physical Review Letters•Oct 23, 2009•16 citations

Origin of entanglement in an H(2𝑝) atom pair produced through the photodissociation of an H2 molecule

Hosaka Koichi, Yutaro Torizuka, Philipp Schmidt +4 more

In this paper, as a continuation of our recent article demonstrating that the 2p atom pair is entangled in the H2 photodissociation [Phys. Rev. A 111, 023116 (2025)], we have discovered the origins of the entanglement for the 2p atom-pair state and obtained the entanglement entropies by origins. The same investigation has been carried out for the precursor molecular state of the 2p atom pair. The reason for the increase of the entanglement entropy during the dissociation from the precursor molecule into the H(2p) + H(2p) atom pair has been made clear.

•Jul 1, 2025•0 citations

Analytical expression for the angular correlation function of two Lyman- α photons in the photodissociation of hydrogen molecules

Kouichi Hosaka, Yutaro Torizuka, Philipp Schmidt +4 more

An analytical expression of the angular correlation function of a pair of Lyman-$\ensuremath{\alpha}$ photons in the photodissociation of a hydrogen molecule is derived theoretically in a manner based on both atomic and molecular physics and quantum optics. The angular correlation function turns out to be expressed in terms of cosine (sine) functions of four angular variables of detectors with five coefficients. The angular correlation function is expanded in terms of the spherical harmonics for investigating which terms are involved, and we discuss the reason why they are involved. Interesting features are revealed in the expansion. We then search for a special detector arrangement where the angular-dependent terms vanish in the angular correlation function expressed in terms of the spherical harmonics. It turns out that no such detector arrangement in fact exists, but there is a special pair of detector arrangements where the angular-dependent terms vanish in the summation of two values of the angular correlation function expressed in terms of the spherical harmonics. We refer to such a pair of detector arrangements as a magic pair.

Physical Review A•Jun 23, 2021•2 citations
An angular correlation function (ACF) of a pair of Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα photons produced in photodissociation of H2{\mathrm{H}}_{2}H2​ with linearly polarized incident light is measured on a whole sphere at a 33.66-eV photon energy so that we may distinctly substantiate that the 2p2p2p atom pairs are in the state previously identified by Torizuka et al. as the 2p2p2p atom-pair state [Phys. Rev. A 99, 063426 (2019)], who measured the ACF only on a circle in the plane perpendicular to the incident light beam. The 2p2p2p atom-pair state determined with much less uncertainty than before is investigated in terms of whether or not the state is entangled. We discover that the pair of fragment H(2p)\mathrm{H}(2p)H(2p) atoms is certainly entangled in terms of the electron-electron entanglement, and obtain the entanglement entropy of the 2p2p2p atom-pair state. The similar discussion is made in terms of the electronic spatial-spin entanglement too, where the spatial motion and spin motion of electrons are considered nonidentical partial systems at variance with the former electron-electron entanglement. It turns out that the pair of fragment H(2p2p2p) atoms is entangled in terms of the electronic spatial-spin entanglement as well as the electron-electron entanglement.
The angular correlation functions (ACFs) of a pair of Lyman-$\ensuremath{\alpha}$ photons in photodissociation of H2{\mathrm{H}}_{2}H2​ and D2{\mathrm{D}}_{2}D2​ are measured with linearly polarized incident light at a 33.66-eV incident photon energy in a wider angular range at a narrower angular step with smaller distortion than before [Y. Nakanishi et al., Phys. Rev. A 90, 043405 (2014)] so that we identify the atom-pair state emitting the pair of Lyman-$\ensuremath{\alpha}$ photons and find out whether the atom pair is entangled or not. Searching for reasonable 2p2p2p atom-pair states that reproduce the experimental ACFs to solve the issue, we show that hydrogen molecules are photoexcited to the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state in the Franck-Condon region and then the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state comes to superpose with the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}(2)$ state as the internuclear distance increases to infinity. The superposition is brought about by the spin-orbit coupling, which is effective around infinite internuclear distance because the potential-energy curves of those states are close to each other, but is negligibly small around the Franck-Condon region because they are apart from each other. The 2p2p2p atom pairs turn out to be in the 1u{1}_{u}1u​ superposition state, which is entangled. We therefore conclude that an entangled pair of hydrogen atoms is spontaneously produced through the photodissociation of a hydrogen molecule, and the entanglement originates from the 1u{1}_{u}1u​ symmetry properties, which are invariant during the dissociation from the Franck-Condon region towards infinite internuclear distance.
The angular correlation functions (ACFs) of a pair of Lyman-\ensuremath{\alpha} photons emitted from H fragments in the photodissociation of a hydrogen molecule are measured at a 33.66-eV incident photon energy and at hydrogen gas pressures of approximately 0.1 and 1 Pa. The ACFs are measured for both opposite and nonopposite arrangements of the two photon detectors. It turns out that the experimental ACFs involve neither the contribution of the reactions H(n=2)+H2\mathrm{H}(n\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}2)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{2}H(n=2)+H2​ nor the contribution of the cascade from H(n\mathrm{H}(nH(n \ensuremath{\geqslant} 3) to H(2p)2p)2p) fragments. Thus the experimental ACFs are those for primary H(2p)2p)2p) pairs following the photodissociation of H2{\mathrm{H}}_{2}H2​. The experimental ACFs are compared with (i) the theoretical ACF for entangled pairs of H(2p)2p)2p) atoms, where the magnetic quantum number of each hydrogen atom is indefinite, and (ii) the theoretical ACF for H(2p)2p)2p) pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis [the former entangled state of H(2p)2p)2p) pairs is a sum of the latter pair states with definite magnetic quantum number]. In the theoretical ACF in (ii), the disentanglement in H(2p)2p)2p) pairs during the dissociation is considered. The experimental ACFs show a similar tendency in angular dependence to the theoretical ACF for entangled H(2p)2p)2p) pairs. However, there still remains a considerable difference in the variation magnitude between those experimental and theoretical ACFs. The experimental ACFs show the reverse tendency in angular dependence to the theoretical ACF for H(2p)2p)2p) pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis. We thus conclude that the pair of H(2p)2p)2p) atoms in the photodissociation of H2{\mathrm{H}}_{2}H2​ is unlikely to be in the definite states of magnetic quantum number of each hydrogen atom relative to the internuclear axis, i.e., unlikely to be in the components of the entangled state of H(2p)2p)2p) pairs.
We have measured the coincidence time spectra of two Lyman-α photons emitted by a pair of H(2p) atoms in the photodissociation of H 2 at the incident photon energy of 33.66 eV and at the hydrogen gas pressures of 0.40 Pa and 0.02 Pa, from which the angular distributions of two Lyman-α photons have been obtained. The experimental angular distributions indicate the generation of the entangled pair of H(2p) atoms as predicted by the theory of our group (Miyagi et al 2007 J. Phys. B 40 617) and the role of a new kind of reaction, i.e. the reaction of the entangled pair of H(2p) atoms with an H 2 molecule that efficiently changes the entanglement. It has turned out that more entangled pairs of H(2p) atoms survive at 0.02 Pa than at 0.40 Pa. The decay time constant in the coincidence time spectrum at 0.02 Pa is approximately half the lifetime of a single H(2p) atom, 1.60 ns, while the decay time constant at 0.40 Pa is in agreement with the lifetime of a single H(2p) atom. It follows that the decay faster than the lifetime of a single H(2p) atom originates from the entanglement in the pair of H(2p) atoms.
The angular distribution of two Lyman-alpha photons, i.e., the probability density that two Lyman-alpha photons are emitted in given directions, in the photodissociation of a hydrogen molecule have been measured at the hydrogen gas pressures of 0.40 and 0.13 Pa. We have found that the experimental angular distributions seem to approach the theoretical one by our group [J. Phys. B 40, 617 (2007)] with decreasing pressure, which indicates the generation of the entangled pair of H(2p) atoms shown in the theory and the role of the reaction of the entangled pair of H(2p) atoms with an H2 molecule that efficiently changes the entanglement.
In this paper, as a continuation of our recent article demonstrating that the 2p atom pair is entangled in the H2 photodissociation [Phys. Rev. A 111, 023116 (2025)], we have discovered the origins of the entanglement for the 2p atom-pair state and obtained the entanglement entropies by origins. The same investigation has been carried out for the precursor molecular state of the 2p atom pair. The reason for the increase of the entanglement entropy during the dissociation from the precursor molecule into the H(2p) + H(2p) atom pair has been made clear.
An analytical expression of the angular correlation function of a pair of Lyman-$\ensuremath{\alpha}$ photons in the photodissociation of a hydrogen molecule is derived theoretically in a manner based on both atomic and molecular physics and quantum optics. The angular correlation function turns out to be expressed in terms of cosine (sine) functions of four angular variables of detectors with five coefficients. The angular correlation function is expanded in terms of the spherical harmonics for investigating which terms are involved, and we discuss the reason why they are involved. Interesting features are revealed in the expansion. We then search for a special detector arrangement where the angular-dependent terms vanish in the angular correlation function expressed in terms of the spherical harmonics. It turns out that no such detector arrangement in fact exists, but there is a special pair of detector arrangements where the angular-dependent terms vanish in the summation of two values of the angular correlation function expressed in terms of the spherical harmonics. We refer to such a pair of detector arrangements as a magic pair.